# Let A be a 6 X 9 matrix. If Nullity (ATA^TAT) = 2 then Nullity(A) = 2

Question
Matrix transformations
Let A be a 6 X 9 matrix. If Nullity $$\displaystyle{\left({A}{T}{A}^{{T}}{A}{T}\right)}$$ = 2 then Nullity(A) = 2

2021-02-12
It is given that AA is a 6×9 matrix and Nullity $$\displaystyle{\left({A}^{{T}}\right)}={2}$$. Let rank(A)=k. Then we know that rank $$\displaystyle{\left({A}^{{T}}\right)}={k}$$. By Rank-nullity theorem we have
rank(A)+Nullity(A)=9andrank(AT)+Nullity(AT)=6
It follows that
rank $$\displaystyle{\left({A}^{{T}}\right)}={6}−{2}={4} ⟹ {k}={4}$$
Therefore we get
Nullity(A)=9−k=9−4=5
Hence the given statement is False.

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